Seif, S., Mignan, A., Zechar, J., Werner, M., & Wiemer, S. (2017).Estimating ETAS: the effects of truncation, missing data, and modelassumptions. Journal of Geophysical Research: Solid Earth, 122(1),449-469. https://doi.org/10.1002/2016JB012809

Publisher's PDF, also known as Version of record

Link to published version (if available):10.1002/2016JB012809

Link to publication record in Explore Bristol ResearchPDF-document

University of Bristol - Explore Bristol ResearchGeneral rights

This document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/

Estimating ETAS: The effects of truncation, missingdata, and model assumptionsStefanie Seif1 , Arnaud Mignan2, Jeremy Douglas Zechar1, Maximilian Jonas Werner3 ,and Stefan Wiemer1

1Swiss Seismological Service, ETH Zürich, Zürich, Switzerland, 2Institute of Geophysics, ETH Zürich, Zurich, Switzerland,3School of Earth Sciences, University of Bristol, Bristol, UK

Abstract The Epidemic-Type Aftershock Sequence (ETAS) model is widely used to describe theoccurrence of earthquakes in space and time, but there has been little discussion dedicated to thelimits of, and influences on, its estimation. Among the possible influences we emphasize in this article theeffect of the cutoff magnitude, Mcut, above which parameters are estimated; the finite length ofearthquake catalogs; and missing data (e.g., during lively aftershock sequences). We analyze catalogs fromSouthern California and Italy and find that some parameters vary as a function of Mcut due to changingsample size (which affects, e.g., Omori’s c constant) or an intrinsic dependence on Mcut (as Mcut increases,absolute productivity and background rate decrease). We also explore the influence of another form oftruncation—the finite catalog length—that can bias estimators of the branching ratio. Being also afunction of Omori’s p value, the true branching ratio is underestimated by 45% to 5% for 1.05< p< 1.2.Finite sample size affects the variation of the branching ratio estimates. Moreover, we investigate theeffect of missing aftershocks and find that the ETAS productivity parameters (α and K0) and the Omori’s cand p values are significantly changed for Mcut< 3.5. We further find that conventional estimation errorsfor these parameters, inferred from simulations that do not account for aftershock incompleteness, areunderestimated by, on average, a factor of 8.

1. Introduction

The Epidemic-Type Aftershock Sequence (ETAS) model is one of the most widely used statistical models todescribe the temporal (and later on spatial) clustering of seismicity [Ogata, 1988, 1998]. While other modelsto describe seismicity have been proposed over the years [Vere-Jones, 1970, 1978, 2005; Kagan and Knopoff,1987; Console and Murru, 2001; Lippiello et al., 2007; Turcotte et al., 2007], we focus on ETAS because it is com-monly used for research and is being applied in operational earthquake forecasting [Marzocchi and Lombardi,2009; Marzocchi and Murru, 2012; Marzocchi et al., 2014; Rhoades et al., 2015] and time-dependent seismichazard [Gerstenberger et al., 2014; Field et al., 2015].

As its name suggests, ETAS was designed to describe aftershocks, in particular by expressing their rate as afunction of time (temporal model) or as a function of time and space (spatiotemporal model). ETAS reflectsthe fundamental observation that aftershocks tend to cluster near and after a main shock. Themodel consistsof two parts: (1) background events, which occur independently but are often interpreted as being caused bythe same underlying process (e.g., tectonic loading due to plate movement) and (2) aftershocks, which aretriggered by other earthquakes, either background events or previous aftershocks. The independent and trig-gered parts are described by the conditional intensity function λwhich quantifies the earthquake rate at timet and location (x, y) (in case of a spatiotemporal model)

λ t; x; yjHtð Þ ¼ μ x; yð Þ þXj:tj<t

g t � tj ; x � xj; y � yj;Mj

� �(1)

where μ(x, y) [time� 1 �distance� 2] is the spatially heterogeneous background rate and g(t� tj, x� xj, y� yj;Mj) [time� 1 �distance� 2] is the triggering function. The triggering function expresses the influence of pastearthquakes with tj< t or equivalently the history Ht of the seismicity up to time t. It consists of empirical dis-tribution functions, which model the productivity of aftershocks by a parent earthquake and the distributionof aftershocks in time and space. It has the following form [Ogata, 1998]:

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 1

PUBLICATIONSJournal of Geophysical Research: Solid Earth

RESEARCH ARTICLE10.1002/2016JB012809

Key Points:• Missing aftershocks bias theproductivity parameters and Omori’s cand p for cutoff magnitudes M< 3.5

• Missing aftershocks cause greateruncertainty in these parameterestimates

• Branching ratio estimators are biasedunder realistic conditions with a finitecatalog length

Supporting Information:• Supporting Information S1

Correspondence to:S. Seif,stefanie.seif@sed.ethz.ch

Citation:Seif, S., A. Mignan, J. D. Zechar,M. J. Werner, and S. Wiemer (2017),Estimating ETAS: The effects oftruncation, missing data, and modelassumptions, J. Geophys. Res. Solid Earth,121, doi:10.1002/2016JB012809.

Received 8 JAN 2016Accepted 7 DEC 2016Accepted article online 9 DEC 2016

©2016. American Geophysical Union.All Rights Reserved.

g t; x; y;Mð Þ ¼ K0 eα M�Mcutð Þ � c p�1 t þ cð Þ�p p� 1ð Þ � f x; yjMð Þ ; (2)

where the temporal and spatial distributions are normalized. In the common ETAS formulation the spatialdecay f follows a power law with

f x; yjMð Þ ¼ d eγ M�Mcutð Þ� �q�1

=π� x2 þ y2 þ d eγ Mj�Mcutð Þ� ��qq� 1ð Þ; (3)

which shows good performance in Ogata and Zhuang [2006] and Werner et al. [2011].

The triggering function provides information about the stability of the earthquake-generating process. Thebranching ratio n describes a process as stable if n< 1, critical if n=1, and supercritical or explosive ifn> 1. In nature we observe extended earthquake sequences over long periods with stable features, but theycould include temporarily explosive processes that culminate in large earthquakes [Sornette and Helmstetter,2002] and could explain accelerated seismicity in the time before large earthquakes (see review by Mignan[2011]). n describes the average number of aftershocks per earthquake when averaged over all magnitudeswith

n ¼ ∫Mmax

M0K0e

α M�M0ð Þ�p Mð Þ dM ¼K0ββ � α

1� e� β�αð Þ Mmax�M0ð Þ

1� e�β Mmax�M0ð Þ for α≠β

K0β Mmax �M0ð Þ1� e�β Mmax�M0ð Þ for α ¼ β

8>><>>: ; (4)

where p(M) is the probability density function (pdf) of the magnitude distribution, β = log(10)*b the exponentof the magnitude distribution,M0 is the magnitude above which data are assumed to be complete, andMmax

is the maximummagnitude assumed in p(M). There exist more definitions of the branching ratio [Helmstetterand Sornette, 2003] which are biased under certain conditions, and we explore this bias in section 4.

Readers not familiar with the details of the ETAS model, we refer to the supporting information, wherewe extensively review the functional form of ETAS: we characterize and explain each parameter of thespatiotemporal ETAS model and briefly summarize the maximum likelihood estimation (MLE) procedure,through which parameter estimates are obtained. We indicate model assumptions and their potentialconsequences.

The ETAS model is extensively used throughout the literature for various purposes, including: (i) thedescription and quantification of seismic characteristics of different tectonic zones, with the distributionof the aftershocks in space and time. Hence, ETAS serves as a tool to understand the interplay betweentectonic forces and observed seismicity rates [Kagan et al., 2010; Chu et al., 2011]. (ii) The ETAS model playsa key role in null-hypothesis testing, e.g., about the physics of earthquake generation [Ogata and Zhuang,2006]. Since the ETAS model incorporates the stochasticity of the clustering process, it represents a betternull hypothesis than the Poisson hypothesis [Werner, 2008] (note that an alternative null hypothesisconsists in using the negative binomial distribution to represent earthquake clustering [e.g., Mignanet al., 2013], (iii) ETAS is commonly used to prospectively forecast seismicity such as in the Collaboratoryfor the Study of Earthquake Predictability project (e.g., Lombardi and Marzocchi [2010] (Italy); [Werneret al. [2011] (California); and Zhuang [2011] (Japan)) or in a modified form to forecast induced seismicity[Bachmann et al., 2011].

The ETAS model has been developed and applied for almost three decades and is now being used routinelyby the statistical seismology community. However, its parameter estimates are influenced by the model for-mulation and its deviation from the observations. The choice of the model depends on the scientific questionand can be formulated, e.g., with or without time-dependent background rate, anisotropic aftershock trigger-ing, spatially and temporally varying ETAS parameters, finite duration of aftershock triggering, and 3-Ddescription of location. Furthermore, parameter estimates are influenced by data truncation through thechoice of the magnitude cutoff Mcut. We dedicate the present study to several of these points: we discussthe bias introduced by incomplete aftershocks and finite catalog length and the effect of data truncationthrough the choice of Mcut.

The data set can be truncated at any Mcut between the minimum magnitude Mc at which the data arethought to be complete (see review and tutorial on Mc by Mignan and Woessner [2012]) and the maximum

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 2

magnitude above which the ETAS estimation procedure is unstable due to sample size limitations. How thischoice affects parameter estimates is discussed in several studies. Wang et al. [2010a] found that by increas-ing the magnitude threshold of the auxiliary window the branching ratio and p decrease, d increases, and αstays approximately the same. Schoenberg et al. [2010] found that by increasing Mcut in the target window c,d, p, and q change and that the true values of p and q are not recovered. They also observe that the trends ofthe changes depend on the formulation of the ETAS model. We find changes in the parameter estimates of cand d, too, which is counterintuitive at first, as the time and space distributions are independent of magni-tude. In addition to the study of Schoenberg et al. [2010] we provide in section 3 an explanation to theseobservations and analytical formulations for the intrinsic dependence of the background rate and K0 onMcut. Our argumentations are similar to those of Harte [2015] who gives detailed explanations on the originof parameter bias based on the concept of broken and false linkages. Beyond the effects considered by Harte[2015] we investigate the influence of incomplete aftershock sequences on parameter estimates.

Incomplete aftershock sequences occur when Mcut is below the completeness threshold of aftershocksequences which is, immediately after a large earthquake, higher than the threshold of general seismicity(Mc(af)>Mc(bulk)). Incomplete data influence the estimates of the productivity parameters α and K0[Helmstetter et al., 2006; Werner et al., 2011; Hainzl et al., 2013; Omi et al., 2014] and the estimate of c [Utsuet al., 1995; Kagan, 2004; Hainzl, 2016a]. The missing data due to incomplete aftershock sequences representa substantial fraction of the total earthquake data: Kagan [2004] found that following the Landers earthquakein 1992, between 17,500 and 28,000 events aboveM= 2 were missing, which corresponds to roughly 25% ofthe Southern Californian catalog [Hauksson et al., 2012]. An obvious solution to avoid data incompletenesswould be to choose a sufficiently large Mcut. However, the price for neglecting small earthquakes is large:small earthquakes may provide insight into the seismic process [Ebel, 2008;Mignan, 2014] or reveal seismicitycharacteristics more clearly, like the occurrence of foreshocks before large earthquakes [Mignan, 2012a;Schurr et al., 2014]. Moreover, including small earthquakes reduce the bias of ETAS parameter estimatesrelated to sample size [Wang et al., 2010b] and the bias related to the dependence of the parameter estimateson Mcut [Schoenberg et al., 2010; Harte, 2015].

The idea that aftershock incompleteness affects ETAS estimation is accepted among researchers, and thereare various approaches to handle the problem. Kagan [1991] and Hainzl et al. [2008] remove all aftershocksin a certain time window after the main shock. Alternatively, Helmstetter et al. [2006] and Werner et al.[2011] describe the completeness magnitude as a function of the main shock magnitude and time. But theirapproach is limited to a different method of estimating parameters: they optimize the parameters for a 1 dayforecast and the estimates depend on the length of the forecasting window. Omi et al. [2013, 2014] modelaftershock incompleteness for each individual aftershock sequence, rather than for a whole earthquake cat-alog, but they only use the temporal ETAS formulation.

Despite these approaches, little research has been performed to quantify and review the bias of ETAS para-meters that arises from missing data. The aim of this article is to investigate the dependence of ETAS para-meters on Mcut and specifically how missing data in aftershock sequences influence the parameterestimates at differentMcut. We also consider how sample size and temporal edge effects can bias the branch-ing ratio, which characterizes the epidemic nature of the process. In section 2, we describe data used to illus-trate the effects of Mcut and missing aftershocks. Using these data, we explore in section 3 how the ETASparameter estimates depend on Mcut; we also quantify the bias that arises from incomplete aftershocksequences, and we show that the uncertainties of the parameter estimates α, K0, p, and c may be grosslyunderestimated when ignoring the effect of missing data. In section 4, we quantify the bias of branching ratioestimators caused by temporal edge effects.

2. Data

To most clearly see how ETAS parameters depend on Mcut and missing aftershocks, we investigate earth-quake catalogs from Southern California, Italy, and simulations.

2.1. Southern California and Italy Earthquake Catalogs

We use relocated earthquake data from the Southern California catalog between 1981 and 2014 [Haukssonet al., 2012] within a polygon that describes the boundary of the Southern California Seismic Network

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 3

coverage area [Hutton et al., 2010]. For Italy, we analyze the homogenized earthquake catalog by Gasperiniet al. [2013] which contains calibrated earthquake data with respect to the moment magnitude for the years1981–2015 within a polygon that covers themainland of Italy. For the ETAS parameter estimation in SouthernCalifornia we take the first 5 years and data outside the spatial polygon as supporting information. In Italy werestrict the data to a depth of 30 km. Because of a network change in 2005 and therefore an increased detect-ability of small earthquakes we consider only data from 16 April 2005 to 31 December 2015 for the MLE andtake 1 year as well as data outside the spatial polygon as supporting information.

The magnitude of completeness is an important constraint in any ETAS study because MLE assumes com-plete data. We estimateMc(bulk) from the full data sets as the magnitude at which the magnitude distributiondeparts from the G-R law (see review byMignan and Woessner [2012]), more precisely here using the methodof Amorese [2007]. For Southern California and Italy we find Mc(bulk) = 2 with standard deviation SD= 0.1(obtained by bootstrapping) in each case. The MLE of the b value of the G-R law is found b=0.99 and 1.01for Southern California and Italy, respectively, with binnedmagnitudes ΔM=0.1 (Figures 1a and 1b).Mc variesin time especially during large aftershock sequenceswhere thedetection threshold is highest in the early stageof aftershock sequences [Kagan, 2004;Helmstetter et al., 2006;Ogata and Katsura, 2006;Werner et al., 2011;Omiet al., 2013, 2014; Hainzl, 2016b]. The value of max(Mc(t)) strongly depends on the estimation method. Hainzl[2016b] found for sample sizes of 10 a maximum of approximately Mc= 4.5 for the Landers and Hector Minesequences. Other studies found max(Mc (t))≈ 4–4.5 with different methods [Kagan, 2004; Helmstetter et al.,2006]. However, the true maximum value, immediately after the main shock, cannot be estimated due to afinite resolution but is likely higher than 4.0–4.5. To visualize the variation we compute Mc(t) in Figures 2aand 2b with a sliding window approach. The estimated Mc(t) is highly sensitive to the choice of the windowlength which determines the degree of smoothing and using 200 earthquakes per window and an offset of50 earthquakes we find max(Mc(t)) =Mc(af) = 3.5 for Southern California. In the case of Italy (Figure 2b), weobserve a lower Mc(t) starting from April 2005, which can be related to changes in the seismic network[Marzocchi and Lombardi, 2009]. Note thatMc varies in space too [Mignan, 2012b]; however, this is not consid-ered here. In the present study we investigate the role of Mcut with Mc(bulk) ≤Mcut≲Mc(af) on ETAS parameterestimation and do not consider the extreme case Mcut<Mc(bulk), which would require an in-depth investiga-tion of the intrinsic scale of the earthquake detection process [Mignan, 2012b;Mignan and Chen, 2015].

2.2. Synthetic Earthquake Catalogs

To explore ETAS estimation in the case where we know the generating model, we consider catalogs fromsimulations that mimic Southern California. We generate catalogs following the procedure proposed byZhuang [2011] (Figure S1). First, we model the background events. We assume that the number is Poissondistributed with the mean corresponding to the identified background events from the SouthernCalifornia earthquake catalog using a stochastic declustering method [Zhuang et al., 2002], with Mcut = 2.We specify their spatial distribution with their smoothed locations using a Gaussian kernel with a bandwidthof 10 km. Simulated background events are distributed according to the resulting spatial density. The

Figure 1. Analysis of the completeness magnitude Mc(bulk) determined from the full catalog for (a) Southern California(1981–2014) and (b) Italy (2005–2015).

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 4

occurrence times of the background events are sampled from a uniform distribution U(t0, tmax), with t0 = 0and tmax = 33 years (corresponding to the length of the Southern California catalog), as the earthquake ratefollows a stationary Poisson process, and their magnitudes are drawn from the G-R distribution withb=0.99, truncated at Mmax = 7.5 (equation (S4)). We choose Mmax such that the value represents the maxi-mum observed magnitude, which is M= 7.3 (plus an error of 0.2) in the catalog. We choose this value notto represent the maximummagnitude which would be expected to happen in Southern California but to cre-ate synthetics which are close to the observed catalog. Once the background events are simulated, wemodeltheir aftershocks, assuming that the number of triggered aftershocks is Poisson distributed with the meanrate controlled by the productivity. The magnitudes are sampled from the G-R distribution truncated atMmax = 7.5 (equation (S4)), the occurrence times from the modified Omori law (equation (2)), and the loca-tions from the isotropic spatial distribution function f (equation (3)). For second-, third-, and higher-generation aftershocks, the triggering step is repeated for every earthquake. We refer the reader toZhuang and Touati [2015] for technical details of the simulation procedure and a simulation code in R. Thesimulation ends when there are no more potential parents (case n< 1; see equation (4)). In the simulationprocedure, magnitudes, occurrence times, and locations are stochastic, which makes each realization unique.

Following this procedure, we generate 30 synthetic ETAS catalogs with Mcut = 2 using ETAS parameters(Table 1) which are estimated following the iterative approach of Zhuang et al. [2002] where ETAS para-meters and the background rate are obtained simultaneously. We require that α= β to correct for theassumption of an isotropic aftershock distribution or temporally stationary background events (see TextS1.2.1). We do not use the exact estimated value for K0 because it describes a supercritical process. Thesupercritical branching ratio is probably caused by the underestimation of p (see section 3.1 and Harte[2015]). The underestimation of p leads to an overestimation of K0 because of correlation [Schoenberget al., 2010; Harte, 2015]. The overestimation of K0 increases the value of n, because n~ K0 (equation(4)). Because simulations require n< 1 we adjust K0 such that n=0.99. We only keep simulations with anumber of M6+ main shocks and a seismicity rate that are similar to the observed catalog: at least 6M6+main shocks (8 observed), which guarantees a similar fraction of missing aftershocks (next paragraph)and less than 180,000 events (130,000 observed) due to computational limitations.

To investigate the effect of aftershock incompleteness on estimation, we also use a set of modified simulatedcatalogs: from each of the original 30 simulations, we remove the aftershocks of M5+ earthquakes. In parti-cular, we remove aftershocks below the threshold Mc(t, m) =m� 4.5� 0.75 � log10(t) following equation

Figure 2. Temporal variation Mc(t) of the completeness magnitude for (a) Southern California (1981–2014) and (b) Italy(1981–2015) calculated using 200 earthquakes per window and an offset of 50 earthquakes.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 5

(15) of Helmstetter et al. [2006] derived from Southern California wherem is the main shock magnitude and tis the time since it occurred. We apply the removal of aftershocks only to the descendants (first or higherorder) of the main shock to assure not to remove spatially unrelated earthquakes. With this procedure weremove about 10% of the data in each catalog. In what follows we refer to these catalogs as complete simu-lated catalogs and incomplete simulated catalogs, respectively.

3. Influence of Cutoff Magnitude on ETAS Parameter Estimates

In every application of the ETASmodel, the modeler must choose aminimummagnitude of interest,Mcut. It isimportant that Mcut is larger than Mc, at which the data are thought to be complete because ETAS does notaccount for missing data. It is also well known that in the wake of a large event, aftershocks that would nor-mally be detected are missed, that isMc(bulk)<Mc(af) (section 2). Data with magnitudes aboveMc(af) should becomplete and a direct application of the MLE method should be possible. However, to maximize the samplesize, one usually chooses forMcut an estimate ofMc(bulk), knowing that this underestimates the completenessmagnitude in the presence of large aftershock sequences. Here we investigate the effect of theMcut for com-plete catalogs and data incompleteness for Mc(bulk) ≤Mcut≲Mc(af) for incomplete catalogs on the ETASparameter estimates.

We estimate ETAS parameters using the MLE method (see Text S1.3) for Southern California, Italy, and syn-thetic catalogs setting Mcut to each value in the set {2.0, 2.5, 3.0, 3.5, 4.0, 4.5}. For Southern California andItaly we estimate the parameters both with and without the condition that α= β; for the simulated catalogswe do only the latter. The dependence of the parameter estimates onMcut for Southern California and Italy isshown in Figure 3 and listed in Table 2. The parameter estimates obtained from the two sets of synthetic cat-alogs are compared in Figure 4. We also compared the estimated parameters pairwise, to be sure that anyvariation is not due to randomness in the limited data set of 30 catalogs. In the following sections we describethe Mcut dependence of each parameter and compare the estimates from the observations of SouthernCalifornia with the simulations.

3.1. K0 and p

K0 describes themean number of offspring aboveMcut of a parent with themagnitudeMcut. In other words K0intrinsically depends on Mcut: K0 stays constant for α= β and decreases for α≠ β with Mcut (Figure 3, K0 andinset). The dependence of K0 on Mcut can be understood with the following scenario: ConsideringMcut1<Mcut2<Mobs, we expect the number of earthquakes aboveMobs (denoted as N1 and N2 for the corre-sponding cutoff magnitudes), generated by a main shock with magnitude m, to be the same. It holds that

N1

N2¼ K1

K2�e

α m�Mcut1ð Þ

eα m�Mcut2ð Þ�e�β Mobs�Mcut1ð Þ

e�β Mobs�Mcut2ð Þ

1 ¼ K1

K2�e β �αð Þ Mcut1�Mcut2ð Þ

K2 ¼K1�e β �αð Þ Mcut1�Mcut2ð Þ for α≠β

K1 for α ¼ β

(:

(5)

Table 1. ETAS Parameters Used for Creating Synthetic Catalogs With Mcut = 2 and α = β = 2.29a

Parameter α = β = 2.29

K0 0.089 (0.080)α 2.29c (day) 0.011p 1.08d (km2) 0.0019q 1.47γ 2.01bg-rate (events/yr) 1070n 0.99

aThe parameters are estimated from Southern California seismicity (1981–2014, with 5 years history). K0 in parenth-eses is adjusted to obtain a stable process (n = 0.99).

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 6

From equation (5) we expect K0 to be constant for α= β. However, its value changes in both reality (Figure 3,K0) and simulations (Figure 4, K0). The simulation reveals furthermore that K0 is overestimated. The overesti-mation of K0 may be due to its anticorrelation with the underestimated p (Figure 4, p).

p is the exponent of the temporal power law and describes the decay rate of aftershocks. The temporal powerlaw is magnitude independent, so we would expect a constant p for different Mcut. However, we observe adecrease. A decrease of p for large Mcut is also reported in Harte [2015]. He explained the decrease of p asa consequence of the decreasing sample size which shifts the peak of the mass close to t=0 to higher values.We test this hypothesis with simulations, where we simulate a large sample of the temporal distribution func-tion (equation (2), Omori law) and estimate parameters p and c for randomly thinned subsamples. Weobserve the contrary of Harte’s [2015] theory: p stays almost constant with a slight increase with decreasingsample size (Figure 5, p increases <0.5% at a sample size of 20) and moreover p is not correlated with thesmallest waiting time (correlation coefficient = 0.03). In order to find an explanation for the observed contra-diction between Harte’s [2015] theory and our simulations, one fact may play the key role: p is not only deter-mined by the highest density part of the power law around t= 0 but also by the large tail of the distribution,which is less sensitive to sample size. Our investigations suggest that for the estimation of p the weight of thetail is larger than the weight of the highest density part.

Figure 3. Parameter estimates are plotted against Mcut for α = β. Data: Italy (2005–2015), black curve; and Southern California (1981–2014), gray curve. The verticalbars represent 2 times the standard errors which are determined from the approximated inverse Hessian of the log-likelihood function. For Southern California theseerrors can be compared with the simulation based 95% quantiles of synthetic incomplete catalogs (orange, compare Figure 4), which are added to the medianestimates of the observed catalogs. In the case where α is treated as a free parameter all estimates but α and K0 follow the same trends as for α = β. The estimates for αand K0 are shown in the inlet. In order to be able to interpret the Southern Californian estimates, we restrict the y axis only to its range of estimates.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 7

Since the decreasing sample size does not explain the decrease of p, the concept of broken links offers analternative explanation. There, links between parents withM<Mcut and their children are broken and conse-quently wrongly associated with another parent event. Orphaned events, which should be correctly referredto as background events, are more likely associated as direct aftershocks if they happen in the aftershock tailof a large earthquake. This widens the distribution of direct aftershocks, and it leads to a larger tail, which canbe modeled by a smaller p.

Figure 4. Parameter estimates are plotted againstMcut for 30 synthetic earthquake catalogs simulated with parameters from Table 1 (gray) and compared with theparameter estimates where aftershocks are removed (orange). The catalogs are generated as described in section 2.2. The branching ratio n (equation (4)) and ne

(equation (9)) are estimates of the complete catalogs.

Table 2. ETAS Parameters Estimated From the Southern Californian Data (1981–2014, With 5 Years History)a

Set 1 Set 2 Set 3 Set 4

Parameter Mcut = 3.5 α fixed Mcut = 2.5 α fixedα free α free

K0 0.51 0.082 0.68 0.084α 1.27 2.29 1.10 2.29c (day) 0.004 0.008 0.007 0.014p 1.09 1.12 1.09 1.11d (km2) 0.231 0.111 0.016 0.007q 1.59 1.72 1.37 1.90γ 1.31 1.85 1.35 1.49bg-rate 38 45 305 434n 1.13 0.75 1.31 0.95

aParameter Sets 1 to 4 are estimated at Mcut = 3.5 and Mcut = 2.5 and the parameter sets are distinguished betweenα = β and α free.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 8

Although p and K0 converge to the true value for decreasingMcut there remains a small bias (overestimation)at M0. An overestimation of p was also reported in Schoenberg et al. [2010]. We suspect that the overestima-tion depends on the model formulation and perhaps on integration problems of the spatial density function,which is a power law function in our study and in Schoenberg et al. [2010], that cannot be integrated analy-tically over an arbitrary polygon as pointed out by Harte [2015]. Furthermore, the bias of K0 and p at M0 maybe explained by the poor performance of the MLE procedure in a close to critical regime (n~ 1), because wefind no bias for n= 0.32.

We observe that both K0 and p are affected by incomplete aftershock sequences. K0 is overestimated by 70%atMcut = 2 and p is underestimated by 2% with respect to the corresponding mean estimate of the completecatalog. The variation in the parameter estimates, represented by the 95% confidence interval from the simu-lated catalogs, for the incomplete catalogs is about 50 (10) times larger for K0 (p) at Mcut = 2 (Table 3).

3.2. α

From the incomplete simulated catalog (Figure 4, α orange) we see that α is significantly underestimated forsmall Mcut ≤ 3 at a significance level of 0.05 (the 95% confidence bounds between complete and incompleteestimates do not overlap) and its value increases with Mcut toward its true value. The variation in the para-meter estimates is larger for the incomplete than for the complete catalogs. Instead of decreasing withMcut, as expected by Schoenberg et al. [2010] and Wang et al. [2010b], the uncertainty is almost constantand at the smallest Mcut = 2 is 10 times as large as the conventionally calculated variation from a completedata set (Table 3).

In the real catalog where α is a free parameter, the α value decreases by 32% fromMcut = 4 to 2 (see Figure 3, αinset). In the synthetic incomplete catalog α decreases by only 3% in the sameMcut range. Missing data due toaftershock incompleteness can therefore not explain the observed decrease of α with decreasing Mcut. Analternative explanation for the decrease, which is not captured in the synthetic catalogs, is an anisotropicaftershock distribution, which leads to a low α (S1.2.1) [Hainzl et al., 2008] or nonstationary background rates,which are shown to have significant influence on α in Southern California [Hainzl et al., 2013]. Hainzl et al.[2013] also show that the intensity of the bias of α increases with a smaller c value. The bias of α in reality,where c is likely lower than estimated (see section 3.3), is hence expected to be larger than we find.

Figure 5. Boxplot of the % error (e.g., c�cc �100) of the estimated ĉ and p for different sample sizes. We simulated a power law

distribution in time according to equation (2) (Omori law) with a sample size of n = 5000 and estimated parameters c and pfor n = 5000 and randomly thinned samples. We repeated the simulations 1000 times. The median c increases withdecreasing sample size and p is not strongly affected by the sample size.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 9

3.3. c

We observe from the estimates of the real and simulated catalogs that c depends onMcut (Figures 3 and 4, c).To determine whether the dependence arises from the varying sample size, which decreases with increasingMcut, we sample a temporal distribution according to the Omori law in equation (2), with different samplesizes and estimate the relevant parameters c and p (Figure 5). The median c value increases about 30% ata sample size of 20. This can be understood because with decreasing sample size, aftershock sequencesare randomly thinned which reduces the effective mass at times around 0 and this can be modeled byincreasing c [Harte, 2015]. We find further evidence for this explanation in the large correlation between cand the smallest waiting time (correlation coefficient = 0.91). However, we cannot exclude the influence offurther sources, e.g., broken links between parents below Mcut and children, on the c value.

Incomplete simulated catalogs (Figure 4, c orange) reveal that aftershock incompleteness increases the cvalue, where the increase is more prominent in the middle range of Mcut (for Mcut = 2.5, the medians fromthe incomplete and complete simulations differ by 85%). An increase in the c value due to aftershock incom-pleteness was also reported by Utsu et al. [1995], Kagan and Houston [2005], and Hainzl [2016a]. Both theinfluence of the sample size and the aftershock incompleteness on c suggest that c depends on the magni-tudeM of the main shock, because it determines the sample size through the number of aftershocks, and thedegree of incompleteness. However, both influences suggest a contrary dependence of c on M: sample sizesuggests a positive and aftershock incompleteness a negative correlation.

3.4. d, q, γ

d is the spatial equivalent to c, and following the same reasoning as described in section 3.3, it depends onthe sample size and therefore increases with Mcut. A different formulation of the spatial distribution f wouldprobably lead to a different effect [Schoenberg et al., 2010]. d is not affected by incomplete aftershocks(Figure 4, d). q, γ have a slightly increasing/decreasing trend, with increasing Mcut, respectively (Figure 4, q, γ).The increase of q can be explained by the decreasing sample size (Figure 5). q does not decrease withMcut as its equivalent p (Figure 4, p, q), due to false links. In space secondary aftershocks distribute isotropi-cally and not one sided as in time; and hence, a wrong association with a parent event leads to adiffusion-type widening of the distribution which is weaker. q estimated from observations shows the sametrend as the simulations and γ shows variations with no clear trend (Figure 3, q and γ). The parameter esti-mates seem to be affected by another source, not present in the simulations, which could be anisotropicaftershock distribution [Helmstetter et al., 2005; Hainzl et al., 2008] or, more generally, inhomogeneous orinconsistent data.

3.5. n

The true and empirical branching ratios (equation (4) and equation (4) where p(M) is replaced by the empiri-cal magnitude distribution) yield supercritical values for small Mcut when considering the whole SouthernCalifornia seismicity (1981–2014) and for all Mcut when considering the whole Italian seismicity (1981–2015) (not shown in the paper). These supercritical branching ratios are not realistic for extensive catalogsand could be related to data or estimation problems. Data problems include (a) uncertainty of the earthquakesource parameters, (b) heterogeneity of seismicity recordings, and (c) incomplete data. Investigating point (a)in Southern California resulted in no significant difference in n after parameter reestimation with perturbedmagnitudes. Investigating (b), we found that the declustered seismicity is not stationary from 1981 to 2014.After taking only data from 2001 to 2014 (using 5 years of data as history before this period) we still obtain

Table 3. Parameter Variation of Incomplete Data Divided by Complete Dataa

Mcut K0 α c p d q γ n υ

2 51.3 9.7 4.5 9.5 1.4 3.3 1.6 25 1.12.5 11.1 4.3 5.3 4.0 1.2 1.7 1.4 6 1.03 1.7 2.0 4.0 1.4 1.1 1.3 0.9 1.2 1.03.5 0.9 0.9 2.2 1.1 1.0 1.0 1.0 0.8 1.04 1.7 1.0 1.3 1.1 1.0 1.0 1.0 1.7 1.04.5 1.2 1.0 1.1 1.0 1.0 1.0 1.0 1.5 1.1

a1.0 means no differences in variation between incomplete and complete data.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 10

n> 1 forMcut = 2 which could be explained by (c) sinceMc(bulk)<Mc(x,y,t) locally (see section 2). Finally, biasesin the parameter estimates, especially in K0 and p, could lead to supercritical branching ratios (see sections 3.1and 5). p is shown to be underestimated at lowMcut due to aftershock incompleteness and at largeMcut prob-ably due to false links (see Figure 4, p and section 3.1). Its underestimation leads to an overestimation of K0because of their negative correlation. This leads to an overestimation of n because of the positive correlationbetween K0 and n. Further research on the branching ratio would be necessary to specify the origin of n> 1.

3.6. Background Rate ρ(x,y) and Relaxation Parameter υ

It is trivial that the background rate ν � ρ(x, y) [events � area� 1 � time� 1] at any point (x, y) decreases withincreasing Mcut. But the scaling of ETAS background event rate does not follow the G-R law. This is because,in any application, earthquakes that are triggered by events with magnitude less than Mcut will be identifiedas apparent background events if the minimum triggering magnitudeM0<Mcut [Sornette and Werner, 2005].Therefore, the background rate estimated from data withM0<Mcut is only an apparent background rate, and

likewise the branching ratio is an apparent branching ratio. Using the apparent branching ratio, nobsa , definedin Sornette and Werner [2005] we derive the number of apparent background events at a given Mcut

(see Appendix A) as

Ncutbg ¼ Ncut

tot � 1� nobsa � eβ�αð Þ Mmax�Mcutð Þ � 1

e β�αð Þ Mmax�Mobsð Þ � 1

� �

¼ Nobstot �eβ� Mobs�Mcutð Þ� 1� Nobs

tot � Nobsbg

Nobstot

� eβ�αð Þ Mmax�Mcutð Þ � 1

e β�αð Þ Mmax�Mobsð Þ � 1

!: (6)

In Figure 6 we compare the true number of background events, obtained through the known triggering his-tory of ETAS simulations, at varying Mcut with different estimates thereof: estimates based on equation (6),based on the G-R law, and based on the declustering method of Zaliapin et al. [2008]. We find that equation

(6) (knowing theNobsbg andNobs

tot atMobs = 3.5) represents the true background rate well (i.e., within the standard

deviation confidence bounds). The estimate based on the G-R relation overpredicts at Mcut< 3.5 (knowing

Nobsbg at Mobs = 3.5). The Zaliapin declustering slightly underestimates at Mcut> 2 and overestimates at small

Mcut = 2.

υ is the relaxation parameter or weighting constant for the background rate, also called failure rate inConsole and Murru [2001]. It is introduced to assure that the integral of the conditional intensity function(equation (1)) over the target spatiotemporal window equals the number of target earthquakes and ishence not a free parameter. Therefore, the integral of ν � ρ(x, y) over the target area should equal thenumber of background events Nbg. It is ν=Nbg/∫x,yρ(x, y)dxdy. When using stochastic declustering, Nbg isthe sum over all background probabilities. υ increases with Mcut for the simulated catalogs (Figure 4, υ)and the true observations (Figure 3, υ).

4. Bias of Branching Ratio Estimators

The branching ratio is not an ETAS parameter but rather an important property of the process; it has an unam-biguous interpretation and implications for the criticality, and therefore predictability, of earthquakes. Thebranching ratio has the following definitions [Helmstetter and Sornette, 2003]: (i) the average number of after-shocks per earthquake when averaged over all magnitudes, (ii) the fraction of events that are triggered, or (iii)the fraction of aftershocks that are second generation or higher generation. (i) is expressed in equation (4)and (ii) and (iii) are expressed, respectively, as

n ¼ Naf

Ntot; and (7)

n ¼ N2ndþhigheraf

Naf; (8)

where Naf is the number of aftershocks and Ntot the total number of earthquakes. In practice, we can replace

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 11

p(M) in equation (4) with the empirical magnitude distribution, avoiding the stochasticity of the magnitudedistribution, and this yields

ne ¼XN

i¼1K0e

α Mi�M0ð Þ=N (9)

(J. Zhuang, personal communication, 2015). Equation (4) is the true branching ratio of an ETAS process, if oneknows the true value of every relevant parameter (K0, α, β,M0,Mmax), while equations (7–9) are estimators thatone applies to real seismicity catalogs via declustering or applies to simulated catalogs, where the triggeringhistory (i.e., all connections between parents and children) is known. Equation (4) also becomes an estimatorin case it is calculated with parameter estimates.

The estimators of the branching ratio in equations 7–9 have never been explored for their consistency. Wewant to explore the catalogs’ temporal edge effects and finite sample properties to detect their influenceon possible bias and variation of the branching ratio estimators in equations 7–9. To focus only on the biasintroduced by limited sample size and finite duration of the catalog, we assume that Naf and Ntot are knownthrough the triggering history of the simulations, such that we can rule out any bias that is induced fromother unmet model assumptions, e.g., finite aftershock duration. By temporal edge effects, we mean effectsarising from the fact that any catalog we analyze (i) will include events triggered by earthquakes occurringbefore the beginning of the catalog and (ii) will miss events triggered after the end of the catalog. We inves-tigate the influence of the finite sample size as a function of the productivity parameters and the memorylength, determined by the size of p and the number of events in a catalog. Those were also found to besources of influence in a related investigation, carried out by Sornette and Utkin [2009], who evaluated theperformance of different declustering methods by comparing n (equation (7)) obtained from declustering

Figure 6. Number of background events (True) of 30 synthetic catalog generated with parameters from Table 1, which areobtained from the triggering history, is compared with the estimated number of background events (Est) obtained withequation (15), given Nobs

bg and Nobstot at Mobs = 3.5. Nobs

bg is the exact number of background events at Mobs. True is alsocompared with the Gutenberg-Richter (G-R) prediction usingNobs

bg and with the number of background events obtained bythe Zaliapin et al. [2008] declustering (Zal).

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 12

techniques with n from synthetic catalogs. For various values of α, K0, and p, we generate 100 synthetic ETAScatalogs with at least 1 million events and apply each of the estimators. Note that because we vary the pro-ductivity parameters, we also implicitly vary the true branching ratio. To consider how the estimates vary as afunction of sample size, we subsample the 100 synthetic catalogs, starting at the end of the catalog andgrowing backward. For example, to estimate bias from catalogs containing 1000 events, we cull the final1000 events of each catalog; we start from the end of the catalog so we have the most history available.

In Figures 7 and 8 we show the bias and spread of the estimators of equations (7) and (8) and equation (9), asa function of sample size for different α, K0, and p. We find that the variation of the estimates in equations (7)and (8) increases as α increases (compare the shaded regions in the first three columns of any row in Figure 7)and decreases with sample size (Figure 7, any plot). The variation of both estimates is almost the same butslightly larger for n of equation (8) (Figure 7, gray versus orange shaded regions). Under realistic conditions(1.05< p< 1.2; Figure 7, first and second rows) n does not converge to the true n. The size of the bias dependson p (different rows in Figure 7). If p is unrealistically large (Figure 7, third row), equations (7) and (8) seem tobe consistent estimators, and when combining a large p with a small α (Figure 7, third row and first and sec-ond columns), the bias is small even for small sample sizes.

Figure 7. Bias of the observed branching ratios n (equations (7) and (8)) from the true n (equation (4)). n is determined for different (first–fourth columns) productivityparameters, (first–third rows) p, and number of events (x axis). One hundred synthetic catalogs are simulated (parameters from Table 1) over 100 years withminimum1,000,000 events in total.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 13

We intend to show now that the observed bias is caused by temporal edge effects and the effect of p on thememory length. Consider a catalog divided into an auxiliary window lasting from t0 to t1, a target windowfrom t1 to t2, and a posttarget window from t2 to∞, with t0< t1< t2. Equations (7) and (8) are calculated usingevents from the target window, and assume we know the complete triggering history, so we know if eachevent in the target window is triggered or background. Certainly, the target window contains some eventstriggered by parents in the auxiliary window—call these incoming aftershocks—and some aftershocks fromparents in the target windowwill fall in the posttarget window—call these outgoing aftershocks. For an eventthat occurs in the target window at time tx, we denote by fout(tx) the fraction of its aftershocks that are out-going. It is described by integration of the modified Omori law (pdf)

g t � txð Þ ¼ p� 1c

1þ t � txc

� ��p; (10)

with

f out txð Þ ¼ ∫∞

t2g t � txð Þdt ¼ 1þ t2 � tx

c

� �1�p

: (11)

Figure 8. Bias of the empirical branching ratio ne from the true n (equations (9) and (4)). ne is determined for different (first–fourth columns) productivity parameters,(first–third rows) p, and number of events (x axis). One hundred synthetic catalogs are simulated (parameters from Table 1) over 100 years with minimum 1,000,000events in total.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 14

The average fraction of outgoing aftershocks per time (normalized by the length of the target window) is

Fout ¼ ∫t2

t1

f out txð Þdtxt2 � t1

¼ c�2þ p

1� c � t1 þ t2c

� �2�p !

1t2 � t1

: (12)

For an event that occurs at ty in the auxiliary window, the fraction of its aftershocks that are incoming is

f in ty� � ¼ ∫

t2

t1g t � ty� �

dt: (13)

Likewise, the average fraction of incoming aftershocks per time (normalized by the length of the targetwindow) is

F in ¼ ∫t1

t0f in ty� �

dty1

t2 � t1

¼ c�2þ p

1� c � t0 þ t1c

� �2�p

þ c � t0 þ t2c

� �2�p

� c � t1 þ t2c

� �2�p !

1t2 � t1

: (14)

If Fin = Fout, the branching ratio estimators are unbiased. In reality, with finite catalog lengths, the equation isnot balanced, resulting in

Fbias p; c; t0; t1; t2ð Þ ¼ Fout � F in ¼ cp�1

�2þ pc þ t1 � t0ð Þ2�p � c þ t2 � t0ð Þ2�p

� �= t2 � t1ð Þ; (15)

and the bias is expressed with

n � n ¼�n�Fbias: (16)

For t2→∞ equation (15) converges (very slowly) to 0, which means that for realistic conditions with finite t2the incoming aftershocks cannot compensate for the outgoing aftershocks. The effect is stronger for smallerp, because the decay rate of aftershocks is lower and hence the memory longer. In Figure 7 we plot� n � Fbiasfor the approximate conditions at a sample size of 0.5 million events: t0 = 0, t1 = 50, and t2 = 100 years and weobserve that it describes the deviation of the estimators from the true value well. When using one of thebranching ratio estimators (equations (7) and (8)) one should be aware of the bias and we recommend cor-recting them with equation (16). We expect the same dependence of the bias on q in the spatial domain.However, the value of q has a larger variability (Table S1), and for large q the bias should be smaller.

For small sample sizes ne consistently underestimates n, and this bias increases with n and α (Figure 8, differ-ent columns in the same row). The reason is that for large α most aftershocks are produced by large events(see S1.2.1), so ne strongly depends on the frequency and the exact magnitude of large events, which are sub-ject to the stochasticity of the G-R law and not completely sampled (especially for large magnitudes) for finiteprocesses. The bias of ne decreases for larger catalogs and converges toward 0 (Figure 8, x axis within oneplot)—in other words, it appears to be a consistent estimator despite temporal edge effects.

5. Discussion

We find that ETAS parameter estimates depend on Mcut, and we detect several reasons for this dependence.Sample size changes the clustering that is observed: as Mcut increases, aftershock clusters are randomlythinned and the intensity of their clustering decreases. The effect of this thinning is primarily seen in the para-meters c and d of the temporal and spatial power laws, respectively, and to a smaller extent in q whichincreases for increasingMcut (Figure 4, c, d, and q). The dependence of c on the sample size argues for a mainshock magnitude dependent c(M), because the aftershock productivity is dictated by M. The exponent q isless sensitive to sample size because it is determined by the decay of the heavy tail of the distribution, andthe median changes by a factor less than 0.05 (Figure 4, q).

The sample size also affects the background rate: it decreases with higher Mcut not according to the G-R lawbut according to equation (6) which accounts for apparent background seismicity [Sornette andWerner, 2005]

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 15

and which is also found by Harte [2015] who additionally investigated second-order biases. K0 directlydepends on Mcut for α≠ β and follows equation (5).

p shows the opposite behavior of its spatial equivalent: it decreases with Mcut. Its behavior can be explainedby broken links between parents withM<Mcut and their children. Rather than being assigned as backgroundevents those orphaned children are more likely assigned as direct aftershocks if they happen in the tail of anaftershock sequence. This widens the distribution of direct aftershocks, which can be modeled by a smaller p.

The same trends for c and d have also been observed for Schoenberg et al. [2010] model 2 (γ=0); however, formodel 1 (γ= α) the trend for d is reverse, which indicates that the trends might be subject to the ETAS formu-lation. They observe a bias for p and q at the lowest Mcut, which we also observe for p. We think that in ourcase the overestimation could result from the model formulation or poor performance of the MLE in a closeto critical regime. Parameters α and K0 are not specifically discussed in Schoenberg et al. [2010]. Our findingsagree with Wang et al. [2010a]: they find a decrease of p and an increase of d with increasing Mcut, α staysconstant. They do not find significant variations for the remaining parameters. Harte [2015] analyzes the biasin the parameter estimates introduced byMcut more theoretically. He finds the same trends for c, d, p, K0, andthe background rate and expresses the dependence of the background rate and K0 similarly to us but using adifferent theoretical concept, namely on broken linkages between parents and children. In agreement to ourresults he finds a strong dependence of p onMcut, which he, in contrast to our findings, leads back to the sam-ple size. His p and K0 estimates are unbiased atM0, whereas ours are slightly overestimated (p overestimationalso observed in Schoenberg et al. [2010]).

ETAS parameter estimates also depend on Mcut because of incomplete data after large earthquakes: thisincompleteness affects the estimates of α, K0, c, and p (Figure 4, orange). We compare the parameter esti-mates from complete synthetic catalogs with estimates from more realistic incomplete synthetic catalogs.We find that the influence of aftershock incompleteness is not significant forMcut ≥ 3.5. This can be explainedby the decreasing fraction of missing data for increasingMcut: whenMcut = 4 the data should be almost com-plete, which is supported by very similar estimates of α, K0, c, and p from complete and incomplete simulatedcatalogs. Only for Mcut ≤ 3 (Mcut ≤ 2.5, 2) do we find a significant difference in α (K0, c, and p) between com-plete and incomplete catalogs. The median α estimate atMcut = 2 from the incomplete catalogs is 7% smallerthan from the complete catalogs. A low α reduces the aftershock productivity of a large main shock anddescribes the incomplete data immediately after the main shock better. Furthermore, K0 estimated fromincomplete catalogs atMcut = 2 is 70% larger than the estimate from complete catalogs. The high K0 compen-sates for the low α. c at Mcut = 2.5 (Mcut = 2) is 85% (75%) larger which supports, e.g., Kagan’s [2004], Hainzl’s[2016a], and Mignan’s [2016b] claim that c could be an artifact of aftershock incompleteness, which reducesthe sample size. And because the intensity of the aftershock incompleteness is determined by themain shockmagnitudeM, it should be considered to model c as a function ofM. p is also influenced by incomplete after-shock sequences and is underestimated by 2% at Mcut = 2.

Within the MLE of incomplete simulated catalogs, α decreases by 3% betweenMcut = 4 andMcut = 2 (Figure 4,α orange). This decrease is much smaller than in the observed catalog, where we also have to deal with miss-ing data, and where we observe a decrease of 32% in this magnitude range (Figure 3, α inset). The higherdecrease of α in the observed catalog could be due to the anisotropic distribution of aftershocks [Hainzlet al., 2008] or time-dependent background rate [Hainzl et al., 2013]. They found that α is systematicallyunderestimated when including the spatial component in the inversion or when assuming a temporal sta-tionary background rate. An estimate of the exact value of α is given in other studies which similarly excludedspatial information and found α≈ β [Felzer, 2004; Helmstetter et al., 2005] and furthermore that Bath’s law canbe reproduced with this assumption [Felzer, 2002]. Also, simulations show that assuming a large α describesobserved seismicity well, but only when missing data are modeled in the simulations too [Helmstetteret al., 2006].

The variations of the parameter estimates for the synthetic catalogs, depicted by the shaded 95% confidenceregions in Figure 4, show large differences between complete and incomplete catalogs for α, K0, c, and p. Asexpected, in complete catalogs the spread of the estimates, indicated by the width of the shaded region, issmallest when the sample size is largest, i.e., when Mcut is smallest [see also Schoenberg et al., 2010; Wanget al., 2010b]. For the incomplete catalogs, the variations of α, K0, and p show the opposite trend—they

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 16

increase with decreasing Mcut where more data are missing. The variation of c does not increase, probablybecause the influence of the varying sample size due to Mcut is larger. However, the variation of c estimatedfrom incomplete catalogs is larger compared to complete catalogs (Table 3). The variation increases becausewe describe incomplete data with a model that assumes complete data. The intensity of incompleteness isvarying for different data sets and depends strongly on the frequency of large (M6+) earthquakes and onthe magnitude of these large earthquakes. Data sets with many large earthquakes are more incompleteand should result in a lower α than data sets with few large earthquakes. Fitting a complete data set doesnot have this dependency and hence a lower variation. For the variation and the estimate itself, we expectthat their degree of dependence on Mcut changes with the chosen data sets and the underlying model(i.e., the choice of the spatial formulation), but trends should remain similar. The proposed analysis couldbe used to estimate the dependences accordingly.

The fluctuations of the observed parameter estimates with Mcut is large and cannot be explained by theiruncertainty estimates based on simulations with aftershock incomplete data (Figure 4). This indicates thatin reality (1) data inhomogeneities could be present, e.g., the completeness could generally vary in spaceand time, (2) the assumptions of a point process model (i.e., earthquakes are points in space and time) arenot met, and (3) the chosen model formulation is a poor representation of nature that does not account for,e.g., time-dependent background rate, anisotropic triggering, finite duration of triggering, temporal and spa-tial varying ETAS parameters, or 3-D earthquake locations.

In this study wewish to point out effects on the two-dimensional representation of the ETASmodel. However,we acknowledge that the three-dimensional effects require a more complex spatial description of the seis-mogenic layer which may lead to different parameter estimates [Guo et al., 2015]. Thus, further investigationof the three-dimensional space is justified but beyond the scope of this paper.

6. Conclusion

There exist different definitions of the branching ratio, which distinguish between estimates (obtained by sin-gle realizations of the point process) and the true branching ratio, obtained by the parameters of the pointprocess which define the mean number of aftershocks per earthquake when averaged over all magnitudes.We found that the properties of the branching ratio estimators n (equations (7) and (8)) depend on the sam-ple size and the temporal finiteness of the catalog. Specifically, a large sample size reduces the variation ofthe estimates, and the temporal finiteness introduces a bias of the estimates, even if the process is sampledsufficiently (1 million events). We found that under realistic conditions, with catalog lengths smaller than sev-eral thousand years and p values between 1.05 and 1.2, n is inconsistent. We provide an analytical formulationto estimate its bias in equation (16) as a function of catalog length, p and c, under the assumption of a suffi-ciently sampled catalog. Our findings show that with perfect knowledge of the triggering structure, one is notable to recover the true n. Consequently, declustering methods, which aim to recover the triggering struc-ture, would always suffer from the bias in n if they showed perfect performance. We described the temporaldecay with the standard modified Omori formulation; for different formulations, e.g., a truncated power lawdecay [Hainzl et al., 2016] or stretched exponential [Mignan, 2015, 2016a, 2016b], we expect a different size ofthe bias.

From the second part of the paper we conclude that ETAS parameter estimates depend onMcut. The samplesize affects the clustering behavior observable in the parameters c and d of the temporal and spatial powerlaws, which increase with increasing Mcut (Figure 4, c and d). Mcut also affects the background rate (equation(15)) and K0 for α≠ β (equation (14)). We find that ETAS parameter estimates α, K0, c, and p are affected byincomplete data after large earthquakes. Incomplete data affect the productivity because fewer events arerecorded in comparison to complete data. The Omori parameters are affected because incompleteness variesas a function of time. In the presence of incomplete data, α and p are underestimated, while K0 and c are over-estimated. The bias increases for decreasing Mcut and is significant for Mcut ≤ 3. This is supported byHelmstetter et al. [2006], who found that it is important to correct for the incompleteness when forecastingaftershock seismicity in time.

Correcting for the induced bias that we found in our simulations is difficult because the bias depends on theparameters, and we estimate parameters from incomplete data. Our simulations show that aftershock

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 17

incompleteness leads to an underestimated α, and we find this trend mirrored in the estimates from obser-vations. Aftershock incompleteness is thus a third identified source of negative bias of α. The other two arethe assumption of isotropy in the spatial aftershock distribution [Hainzl et al., 2008] and a temporally station-ary background rate [Hainzl et al., 2013]. Together, the three sources may explain why α values estimatedfrom MLE of the ETAS model are smaller than the values obtained from more direct measures of productivity(e.g., α= β). Based on our simulations with α= β, the expected bias from aftershock incompleteness alone isabout 10%. Aftershock anisotropy is presumably a greater source of bias. Tectonic setting influences the ani-sotropy and may thus affect the bias, too. If reasons exist to assume α= β, such as those listed at the end ofsection S1.2.1, then α should be fixed during the MLE. In that case, the corresponding K0 that results fromMLEis probably determined by its correlation to α and to a lower degree influenced by the bias. Correcting c and pfor the effect of aftershock incompleteness is not straightforward because the bias likely depends on theother parameters of the process.

We also conclude that the standard deviations of α, K0, c, and p from synthetic incomplete data increase withdecreasingMcut and are orders of magnitudes larger than the commonly estimated standard deviations fromcomplete data forMcut ≤ 3 (Table 3). Therefore, when applying the commonly usedmethods to estimate stan-dard deviations the uncertainties of α, K0, c, and p are underestimated. We recommend to estimate the uncer-tainties for these parameters at Mcut ≤ 3 from simulations which incorporate incompleteaftershock sequences.

This study demonstrates that missing data have a significant effect on the likelihood inference of parametersat low Mcut ≤ 3 and hence on the conclusions that are drawn from the data. We infer that missing data onsmaller and more complex scales in time and space may influence the parameter estimates too, and thisinfluence should also be investigated. The sources of influence on parameter estimates investigated in thispaper (cutoff magnitude, aftershock incompleteness, finite catalog length) only form a subset of all potentialsources of influence, but we show that they have to be considered when drawing conclusions about seismi-city from parameter estimates of different regions or different Mcut.

Appendix A

We derive the relationship between the number of apparent background events and Mcut expressed byequation (6).

Using equation (4) na can be expressed with

na ¼ ∫Mmax

McutK0e

α M�M0ð Þ�p Mð Þ dM ¼n

e β �αð Þ Mmax�Mcutð Þ � 1e β �αð Þ Mmax�M0ð Þ � 1

� �for α≠β

nMmax �Mcut

Mmax �M0

� �for α ¼ β

8>>><>>>:

: (A1)

The dependence of the apparent background rate onMcut is derived in the following. First, the total numberof events at Mcut is estimated according to the G-R law.

Ncuttotal ¼ Nobs

total�eβ Mobs�Mcutð Þ (A2)

where the total number of events at magnitude Mobs is known. Then the number of background events isestimated through equation (7) as

Ncutbg ¼ Ncut

total� 1� ncuta

� �(A3)

where ncuta is given with equation (A1) as

ncuta ¼ nobsae β�αð Þ Mmax�Mcutð Þ � 1e β�αð Þ Mmax�Mobsð Þ � 1

� �: (A4)

Inserting equation (A4) in (A3) the total number of background events is

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 18

Ncutbg ¼ Ncut

total� 1� nobsae β�αð Þ Mmax�Mcutð Þ � 1e β�αð Þ Mmax�Mobsð Þ � 1

� �

¼ Nobstotale

β Mobs�Mcutð Þ� 1� Nobstot � Nobs

bg

Nobstot

e β�αð Þ Mmax�Mcutð Þ � 1e β�αð Þ Mmax�Mobsð Þ � 1

!: (A5)

ReferencesAmorese, D. (2007), Applying a change-point detection method on frequency-magnitude distributions, Bull. Seismol. Soc. Am., 97(5),

1742–1749, doi:10.1785/0120060181.Bachmann, C. E., S. Wiemer, J. Woessner, and S. Hainzl (2011), Statistical analysis of the induced Basel 2006 earthquake sequence: Introducing

a probability-based monitoring approach for enhanced geothermal systems, Geophys. J. Int., 186(2), 793–807, doi:10.1111/j.1365-246X.2011.05068.x.

Chu, A., F. P. Schoenberg, P. Bird, D. D. Jackson, and Y. Y. Kagan (2011), Comparison of ETAS parameter estimates across different globaltectonic zones, Bull. Seismol. Soc. Am., 101(5), 2323–2339, doi:10.1785/0120100115.

Console, R. (2003), Refining earthquake clustering models, J. Geophys. Res., 108(B10), 2468, doi:10.1029/2002JB002130.Console, R., and M. Murru (2001), A simple and testable model for earthquake clustering, J. Geophys. Res., 106, 8699–8711, doi:10.1029/

2000JB900269.Ebel, J. E. (2008), The importance of small earthquakes, Seismol. Res. Lett., 79(4), 491–493, doi:10.1785/gssrl.79.4.491.Felzer, K. R. (2002), Triggering of the 1999 MW 7.1 Hector Mine earthquake by aftershocks of the 1992Mw 7.3 Landers earthquake,

J. Geophys. Res., 107(B9), 2190, doi:10.1029/2001JB000911.Felzer, K. R. (2004), A common origin for aftershocks, foreshocks, and multiplets, Bull. Seismol. Soc. Am., 94(1), 88–98, doi:10.1785/

0120030069.Field, E. H., et al. (2015), UCERF3-ETAS including spatiotemporal clustering for a California Operational Earthquake Forecast (OEF), Oral

Session Presented at the StatSei 9 Meeting, Potsdam, Germany.Gasperini, P., B. Lolli, and G. Vannucci (2013), Empirical calibration of local magnitude data sets versus moment magnitude in Italy, Bull.

Seismol. Soc. Am., 103(4), 2227–2246, doi:10.1785/0120120356.Gerstenberger, M., G. McVerry, D. Rhoades, and M. Stirling (2014), Seismic hazard modeling for the recovery of Christchurch, Earthquake

Spectra, 30(1), 17–29, doi:10.1193/021913EQS037M.Guo, Y., J. Zhuang, and S. Zhou (2015), A hypocentral version of the space–time ETAS model, Geophys. J. Int., 203(1), 366–372, doi:10.1093/gji/

ggv319.Gutenberg, B., and C. F. Richter (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am., 34, 185–188.Hainzl, S. (2016a), Apparent triggering function of aftershocks resulting from rate-dependent incompleteness of earthquake catalogs,

J. Geophys. Res. Solid Earth, 121, 6499–6509, doi:10.1002/2016JB013319.Hainzl, S. (2016b), Rate dependent incompleteness of earthquake catalogs, Seismol. Res. Lett., 87(2A), 337–344, doi:10.1785/0220150211.Hainzl, S., and Y. Ogata (2005), Detecting fluid signals in seismicity data through statistical earthquake modeling, J. Geophys. Res., 110,

B05S07, doi:10.1029/2004JB003247.Hainzl, S., A. Christophersen, and B. Enescu (2008), Impact of earthquake rupture extensions on parameter estimations of point-process

models, Bull. Seismol. Soc. Am., 98(4), 2066–2072, doi:10.1785/0120070256.Hainzl, S., O. Zakharova, and D. Marsan (2013), Impact of aseismic transients on the estimation of aftershock productivity parameters, Bull.

Seismol. Soc. Am., 103(3), 1723–1732, doi:10.1785/0120120247.Hainzl, S., A. Christophersen, D. Rhoades, and D. Harte (2016), Statistical estimation of the duration of aftershock sequences, Geophys. J. Int.,

205(2), 1180–1189, doi:10.1093/gji/ggw075.Harte, D. S. (2015), Model parameter estimation bias induced by earthquake magnitude cut-off, Geophys. J. Int., 204(2), 1266–1287,

doi:10.1093/gji/ggv524.Hauksson, E., W. Yang, and P. M. Shearer (2012), Waveform relocated earthquake catalog for Southern California (1981 to June 2011), Bull.

Seismol. Soc. Am., 102(5), 2239–2244, doi:10.1785/0120120010.Helmstetter, A., and D. Sornette (2003), Importance of direct and indirect triggered seismicity in the ETAS model of seismicity, Geophys. Res.

Lett., 30(11), 1576, doi:10.1029/2003GL017670.Helmstetter, A., Y. Y. Kagan, and D. D. Jackson (2005), Importance of small earthquakes for stress transfers and earthquake triggering,

J. Geophys. Res., 110, B05S08, doi:10.1029/2004JB003286.Helmstetter, A., Y. Y. Kagan, and D. D. Jackson (2006), Comparison of short-term and time-independent earthquake forecast models for

Southern California, Bull. Seismol. Soc. Am., 96(1), 90–106, doi:10.1785/0120050067.Hutton, K., J. Woessner, and E. Hauksson (2010), Earthquake monitoring in Southern California for seventy-seven years (1932–2008), Bull.

Seismol. Soc. Am., 100(2), 423–446, doi:10.1785/0120090130.Kagan, Y. Y. (1991), Likelihood analysis of earthquake catalogues, Geophys. J. Int., 106(1), 135–148, doi:10.1111/j.1365-246X.1991.tb04607.x.Kagan, Y. Y. (2004), Short-term properties of earthquake catalogs and models of earthquake source, Bull. Seismol. Soc. Am., 94(4), 1207–1228,

doi:10.1785/012003098.Kagan, Y. Y., and H. Houston (2005), Relation between mainshock rupture process and Omori’s law for aftershock moment release rate,

Geophys. J. Int., 163(3), 1039–1048, doi:10.1111/j.1365-246X.2005.02772.x.Kagan, Y. Y., and D. D. Jackson (2000), Probabilistic forecasting of earthquakes, Geophys. J. Int., 143(2), 438–453, doi:10.1046/j.1365-

246X.2000.01267.x.Kagan, Y. Y., and L. Knopoff (1987), Statistical short-term earthquake prediction, Science, 236, 1563–1567, doi:10.1126/science.236.4808.1563.Kagan, Y. Y., P. Bird, and D. D. Jackson (2010), Earthquake patterns in diverse tectonic zones of the globe, Pure Appl. Geophys., 167(6-7),

721–741, doi:10.1007/s00024-010-0075-3.Lippiello, E., C. Godano, and L. de Arcangelis (2007), Dynamical scaling in branching models for seismicity, Phys. Rev. Lett., 98(9), 098501,

doi:10.1103/PhysRevLett.98.098501.Lippiello, E., F. Giacco, L. D. Arcangelis, W. Marzocchi, and C. Godano (2014), Parameter estimation in the ETAS Model: Approximations and

novel methods, Bull. Seismol. Soc. Am., 104(2), 985–994, doi:10.1785/0120130148.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 19

AcknowledgmentsWe thank Jiancang Zhuang for provid-ing the code for estimating ETAS para-meters and for his assistance withtechnical questions. We also thank threeanonymous reviewers and the AssociateEditor for their helpful reviews. TheFORTRAN code for estimating ETASparameters can be found at http://bemlar.ism.ac.jp/zhuang/software.html.We thank Paolo Gasperini, Barbara Lolli,and Gianfranco Vannucci for providingthe Italian catalog. Earthquake data forSouthern California were obtained fromhttp://scedc.caltech.edu/research-tools/alt-2011-dd-hauksson-yang-shearer.html.

Llenos, A. L., and J. J. McGuire (2011), Detecting aseismic strain transients from seismicity data, J. Geophys. Res., 116, B06305, doi:10.1029/2010JB007537.

Lombardi, A. M., and W. Marzocchi (2010), The ETAS model for daily forecasting of Italian seismicity in the CSEP experiment, Ann. Geophys.,53(3), 155–164, doi:10.4401/ag-4848.

Marzocchi, W., and A. M. Lombardi (2009), Real-time forecasting following a damaging earthquake, Geophys. Res. Lett., 36, L21302,doi:10.1029/2009GL040233.

Marzocchi, W., and M. Murru (2012), Daily earthquake forecasts during the May–June 2012 Emilia earthquake sequence (northern Italy), Ann.Geophys., 55(4), 561–567, doi:10.4401/ag-6161.

Marzocchi, W., A. M. Lombardi, and E. Casarotti (2014), The establishment of an operational earthquake forecasting system in Italy, Seismol.Res. Lett., 85(5), 961–969, doi:10.1785/0220130219.

Mignan, A. (2011), Retrospective on the Accelerating Seismic Release (ASR) hypothesis: Controversy and new horizons, Tectonophysics,505(1-4), 1–16, doi:10.1016/j.tecto.2011.03.010.

Mignan, A. (2012a), Seismicity precursors to large earthquakes unified in a stress accumulation framework, Geophys. Res. Lett., 39, L21308,doi:10.1029/2012GL053946.

Mignan, A. (2012b), Functional shape of the earthquake frequency-magnitude distribution and completeness magnitude, J. Geophys. Res.,117, B08302, doi:10.1029/2012JB009347.

Mignan, A. (2014), The debate on the prognostic value of earthquake foreshocks: A meta-analysis, Sci. Rep., 4, 4099, doi:10.1038/srep04099.Mignan, A. (2015), Modeling aftershocks as a stretched exponential relaxation, Geophys. Res. Lett., 42, 9726–9732, doi:10.1002/

2015GL066232.Mignan, A. (2016a), Revisiting the 1894 Omori aftershock dataset with the stretched exponential function, Seismol. Res. Lett., 87(3), 685–689,

doi:10.1785/0220150230.Mignan, A. (2016b), Reply to “Comment on ‘Revisiting the 1894 Omori aftershock dataset with the stretched exponential function’ by A.

Mignan” by S. Hainzl and A. Christophersen, Seismol. Res. Lett., 87(5), 1134–1137, doi:10.1785/0220160110.Mignan, A., and C.-C. Chen (2015), The spatial scale of detected seismicity, Pure Appl. Geophys., 119, doi:10.1007/s00024-015-1133-7.Mignan, A., and J. Woessner (2012), Understanding seismicity catalogs and their problems: Estimating the magnitude of completeness for

earthquake catalogs, Community Online Resour. Stat. Seismol. Anal., doi:10.5078/corssa-00180805.Mignan, A., C. Jiang, J. D. Zechar, S. Wiemer, Z. Wu, and Z. Huang (2013), Completeness of the mainland China earthquake catalog and

implications for the setup of the China Earthquake Forecast Testing Center, Bull. Seismol. Soc. Am., 103(2A), 845–859, doi:10.1785/0120120052.

Ogata, Y. (1988), Statistical models for earthquake occurrences and residual analysis for point processes, J. Am. Stat. Assoc., 83(401), 9–27.Ogata, Y. (1998), Space-time point-process models for earthquake occurrences, Ann. Inst. Stat. Math., 50(2), 379–402, doi:10.1023/

A:1003403601725.Ogata, Y., and K. Katsura (2006), Immediate and updated forecasting of aftershock hazard, Geophys. Res. Lett., 33, L10305, doi:10.1029/

2006GL025888.Ogata, Y., and J. Zhuang (2006), Space–time ETAS models and an improved extension, Tectonophysics, 413(1-2), 13–23, doi:10.1016/j.

tecto.2005.10.016.Omi, T., Y. Ogata, Y. Hirata, and K. Aihara (2013), Forecasting large aftershocks within one day after the main shock, Sci. Rep., 3(2218), 1–7,

doi:10.1038/srep02218.Omi, T., Y. Ogata, Y. Hirata, and K. Aihara (2014), Estimating the ETAS model from an early aftershock sequence, Geophys. Res. Lett., 41,

850–857, doi:10.1002/2013GL058958.Rhoades, D. A., M. Liukis, A. Christophersen, and M. C. Gerstenberger (2015), Retrospective tests of hybrid operational earthquake forecasting

models for Canterbury, Geophys. J. Int., 204(1), 440–456, doi:10.1093/gji/ggv447.Schoenberg, F. P. (2013), Facilitated estimation of ETAS, Bull. Seismol. Soc. Am., 103(1), 601–605, doi:10.1785/0120120146.Schoenberg, F. P., A. Chu, and A. Veen (2010), On the relationship between lowermagnitude thresholds and bias in epidemic-type aftershock

sequence parameter estimates, J. Geophys. Res., 115, B04309, doi:10.1029/2009JB006387.Schurr, B., et al. (2014), Gradual unlocking of plate boundary controlled initiation of the 2014 Iquique earthquake, Nature, doi:10.1038/

nature13681.Sornette, D., and A. Helmstetter (2002), Occurrence of finite-time singularities in epidemic models of rupture, earthquakes, and starquakes,

Phys. Rev. Lett., 89(15), 158,501, doi:10.1103/PhysRevLett.89.158501.Sornette, D., and S. Utkin (2009), Limits of declustering methods for disentangling exogenous from endogenous events in time series with

foreshocks, main shocks, and aftershocks, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 79(6 Pt 1), 061110, doi:10.1103/PhysRevE.79.061110.

Sornette, D., and M. J. Werner (2005), Apparent clustering and apparent background earthquakes biased by undetected seismicity,J. Geophys. Res., 110, B09303, doi:10.1029/2005JB003621.

Turcotte, D. L., J. R. Holliday, and J. B. Rundle (2007), BASS, an alternative to ETAS, Geophys. Res. Lett., 34, L12303, doi:10.1029/2007GL029696.Utsu, T. (1961), A statistical study on the occurrence of aftershocks, Geophys. Mag., 30(4), 521–605.Utsu, T., Y. Ogata, and R. S. Matsu’ura (1995), The centenary of the Omori formula for a decay law of aftershock activity, J. Phys. Earth, 43(1),

1–33, doi:10.4294/jpe1952.43.1.Van Stiphout, T., J. Zhuang, and D. Marsan (2012), Models and techniques for analyzing seismicity: Seismicity declustering, Community Online

Resour. Stat. Seismol. Anal., doi:10.5078/corssa-52382934.Veen, A., and F. P. Schoenberg (2008), Estimation of space-time branching process models in seismology using an EM-type algorithm, J. Am.

Stat. Assoc., 103(482), 614–624, doi:10.1198/016214508000000148.Vere-Jones, D. (1970), Stochastic models for earthquake occurrence, J. R. Stat. Soc., 32(1), 1–62.Vere-Jones, D. (1978), Earthquake prediction—A statistician’s view, J. Phys. Earth, 26(2), 129–146, doi:10.4294/jpe1952.26.129.Vere-Jones, D. (2005), A class of self-similar random measure, Adv. Appl. Probab., 37(4), 908–914.Wang, Q., D. D. Jackson, and J. Zhuang (2010a), Missing links in earthquake clustering models, Geophys. Res. Lett., 37, L21307, doi:10.1029/

2010GL044858.Wang, Q., F. P. Schoenberg, and D. D. Jackson (2010b), Standard errors of parameter estimates in the ETAS model, Bull. Seismol. Soc. Am.,

100(5A), 1989–2001, doi:10.1785/0120100001.Werner, M. J. (2008), On the fluctuations of seismicity and uncertainties in earthquake catalogs: Implications and methods for hypothesis

testing, PhD thesis, 308 pp., Univ. of Calif.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 20

Werner, M. J., A. Helmstetter, D. D. Jackson, and Y. Y. Kagan (2011), High-resolution long-term and short-term earthquake forecasts forCalifornia, Bull. Seismol. Soc. Am., 101(4), 1630–1648, doi:10.1785/0120090340.

Zaliapin, I., A. Gabrielov, V. Keilis-Borok, and H. Wong (2008), Clustering analysis of seismicity and aftershock identification, Phys. Rev. Lett.,101(1), 018501, doi:10.1103/PhysRevLett.101.018501.

Zhuang, J. (2011), Next-day earthquake forecasts for the Japan region generated by the ETAS model, Earth Planets Space, 63(3), 207–216,doi:10.5047/eps.2010.12.010.

Zhuang, J., and S. Touati (2015), Stochastic simulation of earthquake catalogs, Community Online Resour. Stat. Seismol. Anal., doi:10.5078/corssa-43806322.

Zhuang, J., Y. Ogata, and D. Vere-Jones (2002), Stochastic declustering of space-time earthquake occurrences, J. Am. Stat. Assoc., 97(458),369–380, doi:10.1198/016214502760046925.

Zhuang, J., Y. Ogata, and D. Vere-Jones (2004), Analyzing earthquake clustering features by using stochastic reconstruction, J. Geophys. Res.,109, B05301, doi:10.1029/2003JB002879.

Zhuang, J., D. Harte, M. J. Werner, S. Hainzl, D. Harte, and S. Zhou (2011), Basic models of seismicity: Spatiotemporal models, CommunityOnline Resour. Stat. Seismol. Anal., doi:10.5078/corssa-07487583.

Journal of Geophysical Research: Solid Earth 10.1002/2016JB012809

SEIF ET AL. ETAS: TRUNCATION AND MISSING DATA 21